Harmonic Series

[paulh428]

Hello, I'm just reading my text on harmonic series and I'm just having trouble where they explain the inequalities. For examples:

s₂ = 1 + 1/2 > 1/2 + 1/2 = 2/2

s₄ = s₂ + 1/3 + 1/4 > s₂ + (1/4 + 1/4) = s₂ + 1/2 > 3/2

So my question is, where are they getting the two halves in s₂ and the two quarters in s₄? Or how do they know to compare the partial sums to those numbers?

[nona.m.nona]

...where are they getting the two halves in s₂ and the two quarters in s₄? Or how do they know to compare the partial sums to those numbers?

Somebody noticed that those numbers would work. Other numbers might work just as well. The proof just needs some numbers that work. And these are the numbers that are customarily used.

The proof for a different series would use other methods or other numbers for the inequalities.